Our aim in this paper is to show how two adaptive focusing techniques, Time Reversal (TR) and the Spatio Temporal Inverse Filter (STIF), are related by the Stokes equations linking waves transmitted and reflected through a medium. For that purpose a model experiment has been investigated: a solid plate located between two arrays of transducers. When sending a wave from an array to the other through the plate, multiple transmitted waves are induced. TR and STIF are used to cancel these echoes. The echoes can be suppressed by TR, using the two arrays cavity surrounding the plate. They can also be cancelled by STIF, inverting the transmission operator between the arrays. The STIF achieves echoes cancellation by using only the transmitted fields through the plate, whereas TR also requires the reflected fields. The STIF's strategy is analyzed in light of the Stokes relations: thanks to the reflections in the medium, it is able to simulate a TR cavity with only one array. A mathematical analysis of the matrix expression of the Stokes relations then leads to two iterative ways to invert the transmission operator. Finally, this general technique is applied to a more complex medium: a human skull bone..

Numerical results for the velocity and attenuation of surface wave modes in fully permeable liquid/partially saturated porous solid plane interfaces are reported in a broadband of frequencies . A modified Biot theory of poromechanics is implemented which takes into account the interaction between the gas bubbles and both the liquid and the solid phases of the porous material through acoustic radiation and viscous and thermal dissipation. This model was previously verified by shock wave experiments. In the present paper this formulation is extended to account for grain compressibility. The dependence of the frequency-dependent velocities and attenuation coefficients of the surface modes on the gas saturation is studied. The results show a significant dependence of the velocities and attenuation of the pseudo-Stoneley wave and the pseudo-Rayleigh wave on the liquid saturation in the pores. Maximum values in the attenuation coefficient of the pseudo-Stoneley wave are obtained in the range of frequencies. The attenuation value and the characteristic frequency of this maximum depend on the liquid saturation. In the high-frequency limit, a transition is found between the pseudo-Stoneley wave and a true Stoneley mode. This transition occurs at a typical saturation below which the slow compressional wave propagates faster than the pseudo-Stoneley wave.

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for “resonance modes” or free oscillations regimes. In the present article, a simple case is investigated, which allows one to enlighten the respective interest of different, classical methods, some of them being rather delicate. This case is the one-dimensional propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain and the residue calculus. Then, the method of separation of variables (space and time) leads to a solution depending on the initial conditions. The question of the orthogonality and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or∕and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode.